# Stochastic Gradient Descent for Spectral Embedding with Implicit Orthogonality Constraint

## TL;DR

This paper introduces a scalable stochastic gradient descent algorithm for spectral embedding that bypasses eigendecomposition by reformulating the problem with an implicit orthogonality constraint, enabling efficient large-scale graph clustering.

## Contribution

It presents a novel reformulation of spectral embedding using an orthogonalization matrix, allowing for mini-batch gradient descent without eigendecomposition.

## Key findings

- Significantly faster execution on large graphs.
- Comparable clustering quality to traditional methods.
- Effective on both synthetic and real datasets.

## Abstract

In this paper, we propose a scalable algorithm for spectral embedding. The latter is a standard tool for graph clustering. However, its computational bottleneck is the eigendecomposition of the graph Laplacian matrix, which prevents its application to large-scale graphs. Our contribution consists of reformulating spectral embedding so that it can be solved via stochastic optimization. The idea is to replace the orthogonality constraint with an orthogonalization matrix injected directly into the criterion. As the gradient can be computed through a Cholesky factorization, our reformulation allows us to develop an efficient algorithm based on mini-batch gradient descent. Experimental results, both on synthetic and real data, confirm the efficiency of the proposed method in term of execution speed with respect to similar existing techniques.